10xy^'-7y=2x^4

{ "query": { "display": "$$10xy^{^{\\prime}}-7y=2x^{4}$$", "symbolab_question": "ODE#10xy^{\\prime }-7y=2x^{4}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "FirstLinear", "default": "y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$10xy^{\\prime}\\left(x\\right)-7y=2x^{4}:{\\quad}y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}$$", "input": "10xy^{\\prime}\\left(x\\right)-7y=2x^{4}", "steps": [ { "type": "interim", "title": "Solve linear ODE:$${\\quad}y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}$$", "input": "10xy^{\\prime}\\left(x\\right)-7y=2x^{4}", "steps": [ { "type": "definition", "title": "First order linear Ordinary Differential Equation", "text": "A first order linear ODE has the form of $$y'\\left(x\\right)+p\\left(x\\right)y=q\\left(x\\right)$$" }, { "type": "interim", "title": "Rewrite in the form of a first order linear ODE", "input": "10xy^{\\prime}\\left(x\\right)-7y=2x^{4}", "result": "y^{\\prime}\\left(x\\right)-\\frac{7}{10x}y=\\frac{x^{3}}{5}", "steps": [ { "type": "step", "primary": "Standard form of a first order linear ODE:", "secondary": [ "$$y'\\left(x\\right)+p\\left(x\\right){\\cdot}y=q\\left(x\\right)$$" ] }, { "type": "step", "result": "10xy^{^{\\prime}}\\left(x\\right)-7y=2x^{4}" }, { "type": "step", "primary": "Divide both sides by $$10x$$", "result": "\\frac{10xy^{^{\\prime}}\\left(x\\right)}{10x}-\\frac{7y}{10x}=\\frac{2x^{4}}{10x}" }, { "type": "step", "primary": "Simplify", "result": "y^{^{\\prime}}\\left(x\\right)-\\frac{7y}{10x}=\\frac{x^{3}}{5}" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$p\\left(x\\right)=-\\frac{7}{10x},\\:{\\quad}q\\left(x\\right)=\\frac{x^{3}}{5}$$" ], "result": "y^{^{\\prime}}\\left(x\\right)-\\frac{7}{10x}y=\\frac{x^{3}}{5}" } ], "meta": { "interimType": "Canon First Order ODE 2Eq" } }, { "type": "interim", "title": "Find the integration factor:$${\\quad}μ\\left(x\\right)=\\frac{1}{x^{\\frac{7}{10}}}$$", "steps": [ { "type": "step", "primary": "Find the integrating factor $$\\mu\\left(x\\right)$$, so that: $$\\mu\\left(x\\right){\\cdot}p\\left(x\\right)=\\mu'\\left(x\\right)$$", "result": "μ^{^{\\prime}}\\left(x\\right)=μ\\left(x\\right)p\\left(x\\right)" }, { "type": "step", "primary": "Divide both sides by $$μ\\left(x\\right)$$", "result": "\\frac{μ^{^{\\prime}}\\left(x\\right)}{μ\\left(x\\right)}=\\frac{μ\\left(x\\right)p\\left(x\\right)}{μ\\left(x\\right)}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{μ^{^{\\prime}}\\left(x\\right)}{μ\\left(x\\right)}=p\\left(x\\right)" }, { "type": "step", "primary": "$$\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=\\frac{μ^{\\prime}\\left(x\\right)}{μ\\left(x\\right)}$$", "result": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{^{\\prime}}=p\\left(x\\right)", "meta": { "general_rule": { "extension": "$$\\frac{d}{dx}\\left(\\ln\\left(f\\left(x\\right)\\right)\\right)=\\frac{\\frac{d}{dx}f\\left(x\\right)}{f\\left(x\\right)}$$" } } }, { "type": "step", "primary": "$$p\\left(x\\right)=-\\frac{7}{10x}$$", "result": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{^{\\prime}}=-\\frac{7}{10x}" }, { "type": "interim", "title": "Solve $$\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=-\\frac{7}{10x}:{\\quad}μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{\\frac{7}{10}}}$$", "input": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=-\\frac{7}{10x}", "steps": [ { "type": "step", "primary": "If$${\\quad}f^{^{\\prime}}\\left(x\\right)=g\\left(x\\right){\\quad}$$then$${\\quad}f\\left(x\\right)=\\int{g\\left(x\\right)}dx$$", "result": "\\ln\\left(μ\\left(x\\right)\\right)=\\int\\:-\\frac{7}{10x}dx" }, { "type": "interim", "title": "$$\\int\\:-\\frac{7}{10x}dx=-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}$$", "input": "\\int\\:-\\frac{7}{10x}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-\\frac{7}{10}\\cdot\\:\\int\\:\\frac{1}{x}dx" }, { "type": "step", "primary": "Use the common integral:$$\\quad\\:\\int\\:\\frac{1}{x}dx=\\ln\\left(x\\right),\\:$$assuming a complex-valued logarithm", "result": "=-\\frac{7}{10}\\ln\\left(x\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "\\ln\\left(μ\\left(x\\right)\\right)=-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}" }, { "type": "interim", "title": "Isolate $$μ\\left(x\\right):{\\quad}μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{\\frac{7}{10}}}$$", "input": "\\ln\\left(μ\\left(x\\right)\\right)=-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}", "steps": [ { "type": "interim", "title": "Apply log rules", "input": "\\ln\\left(μ\\left(x\\right)\\right)=-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}", "result": "μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{\\frac{7}{10}}}", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\ln\\left(μ\\left(x\\right)\\right)=-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}\\quad\\:\\Rightarrow\\:\\quad\\:μ\\left(x\\right)=e^{-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}}$$" ], "result": "μ\\left(x\\right)=e^{-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}}" }, { "type": "interim", "title": "Expand $$e^{-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}}:{\\quad}\\frac{e^{c_{1}}}{x^{\\frac{7}{10}}}$$", "input": "e^{-\\frac{7}{10}\\ln\\left(x\\right)+c_{1}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "result": "=e^{-\\frac{7}{10}\\ln\\left(x\\right)}e^{c_{1}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$e^{-\\frac{7}{10}\\ln\\left(x\\right)}:{\\quad}x^{-\\frac{7}{10}}$$", "input": "e^{-\\frac{7}{10}\\ln\\left(x\\right)}", "result": "=x^{-\\frac{7}{10}}e^{c_{1}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "result": "=\\left(e^{\\ln\\left(x\\right)}\\right)^{-\\frac{7}{10}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "secondary": [ "$$e^{\\ln\\left(x\\right)}=x$$" ], "result": "=x^{-\\frac{7}{10}}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify", "input": "x^{-\\frac{7}{10}}e^{c_{1}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-\\frac{7}{10}}=\\frac{1}{x^{\\frac{7}{10}}}$$" ], "result": "=e^{c_{1}}\\frac{1}{x^{\\frac{7}{10}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:e^{c_{1}}}{x^{\\frac{7}{10}}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{c_{1}}=e^{c_{1}}$$", "result": "=\\frac{e^{c_{1}}}{x^{\\frac{7}{10}}}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } }, { "type": "step", "result": "=\\frac{e^{c_{1}}}{x^{\\frac{7}{10}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7d/R4YQtZG1w27mVetm3alAWQdrNKK9Qvnzg0QLCTxqYGCMMzGhSSXU+Z11oVRwscm8Cy3/+KhUDPMF7zx3LrTDrItGTK5senSF/kdAFuKtRFKk3fejFkyiOiq9iG9IkA2p9s/95F8Tm9GVqHzHZcBKgvgMfM3R6xlR2j4jYL8qkb4EI8bEQ+MqlAeteA0M1y" } }, { "type": "step", "result": "μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{\\frac{7}{10}}}" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xZ/kvw9iKo4JlZMns/8rfxxhaarBteK42nnrae334c/QApBTODfcsWw8XwpOXKFqNvnXrq24Ka6YiPf3rfgIEvY0IJYrOFObS1T90FuhMscnSKF5/4+51qVY0U4KnLmxy0xD7/GKC67MQBjmoJHH4qIvjcm4KMihd3JllH7MbTGYL09W6Dyf/AWTQ8zYjIiCgQUxJPyUNnGfVirkcwpVOz4+ruY5kuligXlC0CtPrr/nGcX57Hy3fYoom/ZxpK5G" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "result": "μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{\\frac{7}{10}}}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{\\frac{7}{10}}}" }, { "type": "step", "primary": "The constant $$e^{c_{1}}\\:$$can be dropped (it will be absorbed into C)", "result": "μ\\left(x\\right)=\\frac{1}{x^{\\frac{7}{10}}}" } ], "meta": { "interimType": "Integrating Factor Top 0Eq" } }, { "type": "interim", "title": "Put the equation in the form $$\\left(\\mu\\left(x\\right){\\cdot}y\\right)'=\\mu\\left(x\\right){\\cdot}q\\left(x\\right):{\\quad}\\left(\\frac{1}{x^{\\frac{7}{10}}}y\\right)^{\\prime}=\\frac{x^{\\frac{23}{10}}}{5}$$", "steps": [ { "type": "step", "primary": "Multiply by the integration factor, $$\\mu\\left(x\\right)$$ and rewrtie the equation as<br/>$$\\left(\\mu\\left(x\\right)\\cdot\\:y\\left(x\\right)\\right)'=\\mu\\:\\left(x\\right)\\cdot\\:q\\left(x\\right)$$", "result": "y^{^{\\prime}}\\left(x\\right)-\\frac{7}{10x}y=\\frac{x^{3}}{5}" }, { "type": "step", "primary": "Multiply both sides by the integrating factor, $$\\frac{1}{x^{\\frac{7}{10}}}$$", "result": "y^{^{\\prime}}\\left(x\\right)\\frac{1}{x^{\\frac{7}{10}}}-\\frac{7}{10x}y\\frac{1}{x^{\\frac{7}{10}}}=\\frac{x^{3}\\frac{1}{x^{\\frac{7}{10}}}}{5}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{x^{\\frac{7}{10}}}-\\frac{7y}{10x^{\\frac{17}{10}}}=\\frac{x^{\\frac{23}{10}}}{5}" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\frac{1}{x^{\\frac{7}{10}}},\\:g=y:{\\quad}\\frac{y^{\\prime}\\left(x\\right)}{x^{\\frac{7}{10}}}-\\frac{7y}{10x^{\\frac{17}{10}}}=\\left(\\frac{1}{x^{\\frac{7}{10}}}y\\right)^{\\prime}$$" ], "result": "\\left(\\frac{1}{x^{\\frac{7}{10}}}y\\right)^{^{\\prime}}=\\frac{x^{\\frac{23}{10}}}{5}" } ], "meta": { "interimType": "Bring Linear To Derivative Form Left 0Eq" } }, { "type": "interim", "title": "Solve $$\\left(\\frac{1}{x^{\\frac{7}{10}}}y\\right)^{\\prime}=\\frac{x^{\\frac{23}{10}}}{5}:{\\quad}y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}$$", "input": "\\left(\\frac{1}{x^{\\frac{7}{10}}}y\\right)^{\\prime}=\\frac{x^{\\frac{23}{10}}}{5}", "steps": [ { "type": "step", "primary": "If$${\\quad}f^{^{\\prime}}\\left(x\\right)=g\\left(x\\right){\\quad}$$then$${\\quad}f\\left(x\\right)=\\int{g\\left(x\\right)}dx$$", "result": "\\frac{1}{x^{\\frac{7}{10}}}y=\\int\\:\\frac{x^{\\frac{23}{10}}}{5}dx" }, { "type": "interim", "title": "$$\\int\\:\\frac{x^{\\frac{23}{10}}}{5}dx=\\frac{2}{33}x^{\\frac{33}{10}}+c_{1}$$", "input": "\\int\\:\\frac{x^{\\frac{23}{10}}}{5}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{5}\\cdot\\:\\int\\:x^{\\frac{23}{10}}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{\\frac{23}{10}}dx", "result": "=\\frac{1}{5}\\cdot\\:\\frac{10}{33}x^{\\frac{33}{10}}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{\\frac{23}{10}+1}}{\\frac{23}{10}+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{\\frac{23}{10}+1}}{\\frac{23}{10}+1}:{\\quad}\\frac{10}{33}x^{\\frac{33}{10}}$$", "input": "\\frac{x^{\\frac{23}{10}+1}}{\\frac{23}{10}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{23}{10}+1:{\\quad}\\frac{33}{10}$$", "input": "\\frac{23}{10}+1", "result": "=\\frac{x^{\\frac{23}{10}+1}}{\\frac{33}{10}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:10}{10}$$", "result": "=\\frac{23}{10}+\\frac{1\\cdot\\:10}{10}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{23+1\\cdot\\:10}{10}" }, { "type": "interim", "title": "$$23+1\\cdot\\:10=33$$", "input": "23+1\\cdot\\:10", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:10=10$$", "result": "=23+10" }, { "type": "step", "primary": "Add the numbers: $$23+10=33$$", "result": "=33" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dlN1ArPOsAu9dDUX+n8eRS061ljBSPJeENOw2efoSWuZHQYndFHh87Qf8kc/VepOpOdmDwqkasIwzm6funut+gezJKvUEKTiF2SMvJoJ/1kkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\frac{33}{10}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "$$x^{\\frac{23}{10}+1}=x^{\\frac{33}{10}}$$", "input": "x^{\\frac{23}{10}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{23}{10}+1:{\\quad}\\frac{33}{10}$$", "input": "\\frac{23}{10}+1", "result": "=x^{\\frac{33}{10}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:10}{10}$$", "result": "=\\frac{23}{10}+\\frac{1\\cdot\\:10}{10}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{23+1\\cdot\\:10}{10}" }, { "type": "interim", "title": "$$23+1\\cdot\\:10=33$$", "input": "23+1\\cdot\\:10", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:10=10$$", "result": "=23+10" }, { "type": "step", "primary": "Add the numbers: $$23+10=33$$", "result": "=33" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dlN1ArPOsAu9dDUX+n8eRS061ljBSPJeENOw2efoSWuZHQYndFHh87Qf8kc/VepOpOdmDwqkasIwzm6funut+gezJKvUEKTiF2SMvJoJ/1kkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\frac{33}{10}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XLT49iOtT5jfBksNzU199c5smC2KlfC9/5bILBjk89Crju+5Z51e/ZZSD3gRHwjBIm/bsE2udYfy/HYoves0chZO2rvLJ1pc7XQlY65dacONC9ASnhnl9Fss0hmPBVXWH9hq2TfsYfTv8Ys/xLhsjXoQu6nS4i4QnngruLyuDs0=" } }, { "type": "step", "result": "=\\frac{x^{\\frac{33}{10}}}{\\frac{33}{10}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{x^{\\frac{33}{10}}\\cdot\\:10}{33}" }, { "type": "step", "result": "=\\frac{10}{33}x^{\\frac{33}{10}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{10}{33}x^{\\frac{33}{10}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7yz6AsKSH1sgsxIvaWEl239OYXPdyEhgTWrx7HQOv/Md7rGjgheXnnNLB7VtiWqH0NrbT4kHr/BM1VbLYTYNi1r484O9NJ/PDg7/ATYMA16bEQEUPeiP7Sq22FVK/Ftxn8QTqIYy6JIhHGkmsubkdVZ6pfF1z6umzUJTJvt+ojYZDOfdBS1fyUJE7OjnCYAFmjR/iZf2csvr9g3oqm1fmec=" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{5}\\cdot\\:\\frac{10}{33}x^{\\frac{33}{10}}:{\\quad}\\frac{2}{33}x^{\\frac{33}{10}}$$", "input": "\\frac{1}{5}\\cdot\\:\\frac{10}{33}x^{\\frac{33}{10}}", "result": "=\\frac{2}{33}x^{\\frac{33}{10}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=\\frac{1\\cdot\\:10x^{\\frac{33}{10}}}{5\\cdot\\:33}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{10x^{\\frac{33}{10}}}{165}" }, { "type": "step", "primary": "Cancel the common factor: $$5$$", "result": "=\\frac{2x^{\\frac{33}{10}}}{33}" }, { "type": "step", "result": "=\\frac{2}{33}x^{\\frac{33}{10}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72XFN01Kv2FoUaXbZQkyUOtTUCkb/sWOYO4oi3qw9wSkdZWIrXDtqc5yeLdFLidQsAbuplQ8eHh51/+y6M1gVwXCQoYlYQ8U+Tfyx0kyzI8hRPTnc5/HL26nHOrjkXBL1dtqWRbKqfO3EZWPFdFz6oP8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUl4OalszaEs+IYQMYhszoNialgGNtOT6ATSAJ3aM5inw54dul0+1tPhOIsSZsq2v18s4GVIyVmgQBarodeOI5aVA==" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{2}{33}x^{\\frac{33}{10}}+c_{1}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "\\frac{1}{x^{\\frac{7}{10}}}y=\\frac{2}{33}x^{\\frac{33}{10}}+c_{1}" }, { "type": "interim", "title": "Isolate $$y:{\\quad}y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}$$", "input": "\\frac{1}{x^{\\frac{7}{10}}}y=\\frac{2}{33}x^{\\frac{33}{10}}+c_{1}", "steps": [ { "type": "interim", "title": "Multiply both sides by $$x^{\\frac{7}{10}}$$", "input": "\\frac{1}{x^{\\frac{7}{10}}}y=\\frac{2}{33}x^{\\frac{33}{10}}+c_{1}", "result": "y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}", "steps": [ { "type": "step", "primary": "Multiply both sides by $$x^{\\frac{7}{10}}$$", "result": "\\frac{1}{x^{\\frac{7}{10}}}yx^{\\frac{7}{10}}=\\frac{2}{33}x^{\\frac{33}{10}}x^{\\frac{7}{10}}+c_{1}x^{\\frac{7}{10}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1}{x^{\\frac{7}{10}}}yx^{\\frac{7}{10}}=\\frac{2}{33}x^{\\frac{33}{10}}x^{\\frac{7}{10}}+c_{1}x^{\\frac{7}{10}}", "result": "y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{1}{x^{\\frac{7}{10}}}yx^{\\frac{7}{10}}:{\\quad}y$$", "input": "\\frac{1}{x^{\\frac{7}{10}}}yx^{\\frac{7}{10}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:yx^{\\frac{7}{10}}}{x^{\\frac{7}{10}}}" }, { "type": "step", "primary": "Cancel the common factor: $$x^{\\frac{7}{10}}$$", "result": "=1\\cdot\\:y" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:y=y$$", "result": "=y" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+/z0awQSODT5Pm74XemsE2c0aKKMD74pRXOcB1ncYZBGHvWcyT9EcJvR94u2LEkGAJYpRu9XpYrd8NSAW2DdD1g26otRrAyNcKKxM/RkFtGjeh7+jKEzLb7VNCEMF3Z/bMzoTd+5nEXVeQoBhpFcIKsUb9eNF/WJwg69dHKwGB/8I9K8KAxxEsxXcZIvmQVoLJnfyopwxoxNhoGcro4dXL8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "interim", "title": "Simplify $$\\frac{2}{33}x^{\\frac{33}{10}}x^{\\frac{7}{10}}+c_{1}x^{\\frac{7}{10}}:{\\quad}\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}$$", "input": "\\frac{2}{33}x^{\\frac{33}{10}}x^{\\frac{7}{10}}+c_{1}x^{\\frac{7}{10}}", "steps": [ { "type": "interim", "title": "$$\\frac{2}{33}x^{\\frac{33}{10}}x^{\\frac{7}{10}}=\\frac{2x^{4}}{33}$$", "input": "\\frac{2}{33}x^{\\frac{33}{10}}x^{\\frac{7}{10}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{\\frac{33}{10}}x^{\\frac{7}{10}}=\\:x^{\\frac{33}{10}+\\frac{7}{10}}$$" ], "result": "=\\frac{2}{33}x^{\\frac{33}{10}+\\frac{7}{10}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2x^{\\frac{33}{10}+\\frac{7}{10}}}{33}" }, { "type": "interim", "title": "$$x^{\\frac{33}{10}+\\frac{7}{10}}=x^{4}$$", "input": "x^{\\frac{33}{10}+\\frac{7}{10}}", "steps": [ { "type": "interim", "title": "Join $$\\frac{33}{10}+\\frac{7}{10}:{\\quad}4$$", "input": "\\frac{33}{10}+\\frac{7}{10}", "result": "=x^{4}", "steps": [ { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{33+7}{10}" }, { "type": "step", "primary": "Add the numbers: $$33+7=40$$", "result": "=\\frac{40}{10}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{40}{10}=4$$", "result": "=4" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72dI+EKvHS4O9ihVKEPUw5w1OdatWgTnqv0M9UtyBXYlFdiKL0bo+0lLRUzNXm1IE690KeCMm6xoofrXTyJV7pHK2/hStT33ECPb9OLmdjQ/+4mT4L5a8t+d0CZdvtlTcM25pkdQGvUJ1S+ZVtBEoLt9t87dd9ayKcHilpWT2C+M=" } }, { "type": "step", "result": "=\\frac{2x^{4}}{33}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7P/jCtAPcRpIx3759tJw/0XbalkWyqnztxGVjxXRc+qC+AaHJwtzM//sSw2fjEnsIRXYii9G6PtJS0VMzV5tSBP2i9gqKNBiEkMJvG7+cA4lpYRIZVxTz+TYWfXQybL4C/z//r+dXk7h9vxeDCLuZqkFSldQSbASvVGIFyDMJQTe50eiE5eSfvyLMLIyJt92ivFvFYunGmv+w716sek9u/Oh1xweLHqTssnV0VruSpv3WNd4gBC3XHrN67a6DCo3r" } }, { "type": "step", "result": "=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7P/jCtAPcRpIx3759tJw/0XbalkWyqnztxGVjxXRc+qC+AaHJwtzM//sSw2fjEnsI8uDBJS7zOB3uC2mldSy10JJfDppZ/3y1BTCxqC3obAcDnzlbPZjyKgy1eUCFsLd5Rvoby4iszK2vZB9pEBr8cpTn0gvMnBku3qkIjjMRzE7N7QDjMsRyRH18ZNWwntaCZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz29YM/VUaVJ7ZBOH9CQOiANdtqWRbKqfO3EZWPFdFz6oL4BocnC3Mz/+xLDZ+MSewjy4MElLvM4He4LaaV1LLXQmdEBP3D49YGy9e62UsciMg==" } }, { "type": "step", "result": "y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}" } ], 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Solve Title 1Eq" } }, { "type": "step", "result": "y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}" } ], "meta": { "interimType": "ODE Solve Linear 0Eq" } }, { "type": "step", "result": "y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "#>#ODE#>#y=\\frac{2x^{4}}{33}+c_{1}x^{\\frac{7}{10}}" } } }, "meta": { "showVerify": true } }